reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem
  a|^2 + b|^2 = c|^2 implies 3 divides a*b*c
  proof
    A0: a|^2*b|^2*c|^2 = (a*b)|^2*c|^2 by NEWTON:7
    .= (a*b*c)|^2 by NEWTON:7;
    assume
    A1: a|^2 + b|^2 = c|^2; then
    per cases by LmAB3;
    suppose
      3 divides a|^2*b|^2*c|^2;
      hence thesis by A0,NAT_3:5,PEPIN:41;
    end;
    suppose
      3 divides a|^2 - b|^2 & not 3 divides a|^2*b|^2*c|^2; then
      B2: not 3 divides (a|^2)*(b|^2*c|^2) & not 3 divides (b|^2)*(a|^2*c|^2)
      & not 3 divides (c|^2)*(b|^2*a|^2); then
      not 3 divides a|^2 & not 3 divides b|^2 & not 3 divides c|^2
        by INT_2:2; then
      not 3 divides a & not 3 divides b & not 3 divides c by Th14; then
      not 3 divides (c*a) by INT_5:7,PEPIN:41; then
      3 divides (c+a)*(c-a) by Th50; then
      3 divides c|^2 - a|^2 by NEWTON01:1;
      hence thesis by A1,B2,INT_2:2;
    end;
  end;
